Paragraph 3.3.1 Traces and K0

Let be a -algebra, then a bounded trace on is a bounded linear map with the trace property: The trace property implies that if and are Murray-von Neumann equivalent, then .
A trace is positive if it maps positive elements to positive elements.
If a trace is positive and then is called a tracial state.

Any trace induces a natural trace on the matrix algebra by taking each
this trace on the matrix algebra then gives rise to a function and this function satisfies Proposition 3.1.8 (Universal property of K0) so there exists a unique group homomorphism which satisfies